Sims-Uhlig (1991) Understanding Unit Rooters: A Helicopter Tour

Summary

In AR(1) models with possible unit roots, the likelihood function is symmetric around the Maximum Likelihood Estimate (MLE) $\hat\rho$ — exactly as in stationary models — even though the MLE's classical sampling distribution is severely right-skewed at $\rho=1$. The paper demonstrates this via a Monte Carlo "helicopter tour" of the three-dimensional (3D) joint probability density function (pdf) surface of $(\rho, \hat\rho)$ sliced from multiple angles. A direct consequence is that flat-prior Bayesian posteriors are symmetric and well-behaved near unit roots, while using Dickey-Fuller (DF) p-values as if they were posterior probabilities implicitly imposes a data-dependent prior that irrationally favors explosive values of $\rho$.

Key Claims

• Likelihood symmetry vs. estimator asymmetry: In AR(1) $y_t = \rho y_{t-1} + \varepsilon_t$, the cross-section of the joint pdf along $\hat\rho = c$ (the posterior under flat prior) is symmetric about $\rho = c$; the cross-section along $\rho = 1$ (the classical sampling distribution) is asymmetric with mass skewed below 1. Same surface, different slices.
• Flat-prior Bayesian inference is valid and simple: Because the likelihood is Gaussian-symmetric in $\rho$, the flat-prior posterior is the same shape as in a static regression. Standard $t$ and $F$ statistics describe the likelihood shape correctly, regardless of whether $\rho = 1$.
• DF p-values as pseudo-posteriors imply an irrational prior: For $\hat\rho = 0.95$, the DF test gives p-value 0.04 for $H_0:\rho=0.9$ (reject) and 0.12 for $H_0:\rho=1$ (fail to reject), yet the posterior probability $P(\rho < 0.9|\hat\rho=0.95) = P(\rho > 1|\hat\rho=0.95) \approx 0.07$ — perfectly symmetric. Using DF p-values as posteriors implicitly requires believing $\rho = 1.05$ is more likely a priori than $\rho = 0.95$.
• The implicit prior is sample-dependent and explosive: Figure 8 shows the prior needed to rationalize DF p-values as posteriors: it shifts weight above $\rho=1$ as $\hat\rho \to 1$, increasing without bound into the explosive region for all observed $\hat\rho$ values. No single prior rationalizes this procedure.
• Practical recommendation: Report conventional (uncorrected) $t$ and $F$ statistics alongside any unit-root asymptotic p-values. The likelihood shape is the same in autoregressions as in regressions on exogenous variables. "The complicated apparatus of classical unit root asymptotics is of little practical value."

Concepts Introduced or Extended

• Unit Root Inference

Entities Mentioned

• Christopher Sims
• Harald Uhlig

Quotes

"We argue that these results imply at a minimum that the usual test statistics and covariance matrices for autoregressions… should be reported without any corrections for the special unit root distribution theory."

"The complicated apparatus of classical unit root asymptotics is of little practical value."

"A $t$ statistic of 3.1 or an $F$ statistic of 1.7 tell us the same thing about the shape of the likelihood in an autoregression as in a regression on exogenous variables."

My Take

The paper's core result — likelihood symmetry despite estimator asymmetry — is mathematically correct and practically important. The helicopter tour visualization is elegant and pedagogically effective. The implicit-prior calculation is a powerful reductio: no sane prior would make $\rho=1.05$ more probable than $\rho=0.95$ a priori. The practical recommendation (report uncorrected $t$ and $F$ statistics) remains controversial among frequentist econometricians who argue that correct size control under the null is what matters for hypothesis testing, but it is fully coherent within the likelihood/Bayesian framework. The paper is essentially a companion piece to Sims (1988) "Bayesian Skepticism on Unit Root Econometrics," here adding graphical demonstrations and the implicit-prior calculation.